A&A 539, A39 (2012) DOI: 10.1051/0004-6361/201118366 & c ESO 2012

Approximations for radiative cooling and heating in the solar

M. Carlsson1,2 and J. Leenaarts1,3

1 Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway e-mail: [email protected] 2 Center of Mathematics for Applications, University of Oslo, PO Box 1053 Blindern, 0316 Oslo, Norway 3 Sterrekundig Instituut, Utrecht University, Postbus 80 000, 3508 TA Utrecht, The Netherlands

Received 31 October 2011 / Accepted 4 January 2012

ABSTRACT

Context. The radiative energy balance in the solar chromosphere is dominated by strong spectral lines that are formed out of LTE. It is computationally prohibitive to solve the full equations of radiative transfer and statistical equilibrium in 3D time dependent MHD simulations. Aims. We look for simple recipes to compute the radiative energy balance in the dominant lines under solar chromospheric conditions. Methods. We use detailed calculations in time-dependent and 2D MHD snapshots to derive empirical formulae for the radiative cooling and heating. Results. The radiative cooling in neutral hydrogen lines and the Lyman continuum, the H and K and intrared triplet lines of singly ionized calcium and the h and k lines of singly ionized magnesium can be written as a product of an optically thin emission (dependent on temperature), an escape probability (dependent on column mass) and an ionization fraction (dependent on temperature). In the cool pockets of the chromosphere the same transitions contribute to the heating of the gas and similar formulae can be derived for these processes. We finally derive a simple recipe for the radiative heating of the chromosphere from incoming coronal . We compare our recipes with the detailed results and comment on the accuracy and applicability of the recipes. Key words. radiative transfer – : chromosphere

1. Introduction chromosphere. We also get heating in the chromosphere from coronal radiation impinging from above. We will here address all The chromospheric radiative energy balance is dominated by three processes: cooling from lines and continua, heating from a small number of strong lines from neutral hydrogen, singly the same transitions and heating through absorption of coronal ionized calcium and singly ionized magnesium (Vernazza et al. radiation. 1981) although a large number of iron lines may be equally We use detailed radiative transfer calculations from dynami- important in the lower (Anderson & Athay 1989). cal snapshots to derive simple lookup tables that only use easily The strong lines are formed out of Local Thermodynamic obtainable quantities from a single snapshot from a simulation. Equilibrium (LTE) which means that numerical simulations of We test these recipes by applying them to several simulation runs the dynamic chromosphere need to solve simultaneously the and compare the radiative losses from detailed solutions with (magneto-)hydrodynamic equations, radiative transfer equations those from the recipes and find in general good agreement. and rate equations for all radiative transitions and energy lev- The structure of the paper is as follows: in Sect. 2 we els involved. This is a major computational effort that has been describe the hydrodynamical snapshots used for deriving the accomplished in one dimensional simulations of wave propa- lookup tables, in Sect. 3 we describe the atomic models used, gation in the quiet solar atmosphere (Klein et al. 1976, 1978; in Sect. 4 we derive the recipes for radiative cooling from strong Carlsson & Stein 1992, 1994, 1995, 1997, 2002; Rammacher & lines of hydrogen, calcium and magnesium, in Sect. 5 we de- Ulmschneider 2003; Fossum & Carlsson 2005, 2006), rive the recipes for heating of cool pockets of gas from the same (Bard & Carlsson 2010), solar flares (Abbett & Hawley 1999; lines, in Sect. 6 we compare the recipes with a detailed calcula- Allred et al. 2005; Kašparová et al. 2009; Cheng et al. 2010)and tion, in Sect. 7 we derive the recipes for the heating by incident stellar flares (Allred et al. 2006). coronal radiation and in Sect. 8 we end with our conclusions. In three-dimensional simulations the computational work re- quired to solve the full set of equations becomes prohibitive; there is a need for a simplified description that retains as much as 2. Atmospheric models possible of the basic physics but brings the computational work To calculate the semi-empirical recipes we use two sets of hy- to tractable levels. The aim of this paper is to provide such sim- drodynamical snapshots. The first is a simulation of wave propa- plified recipes for the chromospheric radiative energy balance. gation in the solar atmosphere using the RADYN code (Carlsson In addition to the cooling in strong lines and continua the & Stein 1992, 1994, 1995, 1997, 2002). RADYN solves the same transitions may provide heating of cool pockets in the one-dimensional equations of conservation of mass, momentum, Article published by EDP Sciences A39, page 1 of 10 A&A 539, A39 (2012) energy and charge together with the non-LTE radiative transfer and population rate equations, implicitly on an adaptive mesh. The grid positions of the adaptive mesh are determined by a grid equation that specifies where high resolution is needed (typically 4p 4p, where there are large gradients). The grid equation is solved si-

8498 8542 multaneously with the other equations. The simulation includes 8662 a 5-level plus continuum hydrogen atom, 5-levels of singly - ized calcium and one level of doubly ionized calcium and a 9-level helium atom including 5 levels of neutral helium, 3 of 3d 3d,

3933 singly ionized helium and one level of doubly ionized helium. 3968 All allowed radiative transitions between these levels are in- cluded. At the bottom boundary we introduce a velocity field that was determined from MDI observations of the Doppler shifts of the 676.78 nm Ni i line. Coronal irradiation of the chromosphere 4s was included using the prescription of Wahlstrom & Carlsson 2S 2P 2D 1S ff (1994) based on Tobiska (1991). The e ects of non-equilibrium ii ionization are handled self-consistently. The simulation is es- Fig. 1. Term diagram of the Ca model atom. The continuum is not shown. Energy levels (horizontal lines), bound-bound transitions (solid) sentially the one used in Judge et al. (2003) but with updated and bound-free transitions (dashed). atomic data for calcium. A number of ingredients that are poten- tially important for the chromospheric energy balance are miss- ing: line blanketing is not included (especially the iron lines may be important, Anderson & Athay 1989), cooling in the strong 3. Atomic models Mg ii h and k lines is not included and the description of partial redistribution is very crude for hydrogen and altogether missing 3.1. Hydrogen for calcium. Most important is probably the restriction of only one dimension. This prevents shocks from spreading out, forces We take the population densities directly from the RADYN sim- shock-merging of overtaking shocks and prevents the very low ulation and thus use the same atomic parameters. We use a five- temperatures found in 2D and 3D simulations from strong ex- level plus continuum model atom with energies from Bashkin pansion in more than one dimension. & Stoner (1975), oscillator strengths from Johnson (1972), pho- toionization cross-sections from Menzel & Pekeris (1935)and The other snapshot is from a 2D simulation of the solar atmo- collisional excitation and ionization rates from Vriens & Smeets sphere spanning from the upper to the corona. (1980) except for Lyman-α where we use Janev et al. (1987). This simulation was performed with the radiation-magneto- We use Voigt profiles with complete redistribution (CRD) for hydrodynamics code Bifrost (Gudiksen et al. 2011) and includes all lines. For the Lyman lines, profiles truncated at six Doppler ff the e ects of non-equilibrium ionization of hydrogen using the widths are used to mimic effects of partial redistribution (PRD) methods described in Leenaarts et al. (2007). A detailed de- (Milkey & Mihalas 1973). scription of the setup and results of this simulation is given in Leenaarts et al. (2011). Also here we have shortcomings in the description. Line blanketing is included but with a two- 3.2. Calcium level formulation for the scattering probability, hydrogen non- We use a five-level plus continuum model atom. It contains the equilibrium ionization is included but the excitation balance ii iii (needed to calculate the individual lines) is very crude. five lowest energy levels of Ca and the Ca continuum (see Fig. 1). The energies are from the NIST database with data com- Both simulations thus have their shortcomings. The purpose ing from Sugar & Corliss (1985), the oscillator strengths are of this paper is, however, to develop recipes that reproduce the from Theodosiou (1989) with lifetimes agreeing well with the results of detailed non-LTE calculations. It is thus not crucial experiments of Jin & Church (1993) and the broadening pa- that the simulations reproduce solar observations in detail as rameters are from the Vienna Atomic Line Database (VALD) long as they cover the parameter space expected in the solar (Piskunov et al. 1995; Kupka et al. 1999). Photoionization chromosphere. cross-sections are from the Opacity project. Collisional excita- For hydrogen it is important to include the coupling be- tion rates are from Meléndez et al. (2007) extrapolated beyond tween the non-equilibrium ionization and the non-LTE excita- the tabulated temperature range of 3000 K to 38 000 K using tion and we therefore use the RADYN simulation for the hydro- Burgess & Tully (1992). Collisional ionization rates from the gen recipes. For magnesium and calcium it is more important ground state are from Arnaud & Rothenflug (1985) and from to include PRD and have a large temperature range than to in- excited states we use the general formula provided by Burgess clude effects of non-equilibrium ionization and we choose the & Chidichimo (1983). Autoionization is treated according to Bifrost 2D simulations. Non-equilibrium ionization is not im- Arnaud & Rothenflug (1985) and dielectronic recombination portant for calcium (Wedemeyer-Böhm & Carlsson 2011) while according to Shull & van Steenberg (1982). The abundance is Rammacher & Ulmschneider (2003) and test-calculations with taken from Asplund et al. (2009). We assumed partial frequency RADYN indicate that ionization/recombination times are as long redistribution (PRD) of radiation over the line profiles for the as 100−500 s for magnesium and therefore of importance for the H&K lines. ionization balance. However, we will see that magnesium is al- ii most exclusively in the form of Mg in the temperature range 3.3. Magnesium where magnesium is important for the cooling/heating of the chromosphere and the neglect of non-equilibrium ionization is We use a three-level plus continuum model atom, including therefore not important for the results. the ground state, the upper levels of the h & k lines and the

A39, page 2 of 10 M. Carlsson and J. Leenaarts: Radiative energy balance recipes continuum. Energy levels and oscillator strengths are from we expect the total collisional excitation from the ground state the NIST database, collisional excitation rates are from Sigut to be equal to the radiative losses: & Pradhan (1995), collisional ionization from Seaton’s semi- Assume a gas where empirical formula (Allen 1964, p. 42) and photoionization cross- 1 sections are from the Opacity Project . The abundance is taken nl from Asplund et al. (2009). The h & k lines are computed assum- R ji C jk, i < j, (2) ing PRD. k=1 − j 1 nl R jk R jk, (3) 4. Cooling from lines and continua k=1 k= j+1 ffi The basic cooling process is an impact excitation fol- here R ji is the radiative rate coe cient from bound level j to i ffi lowed by a radiative deexcitation. If this escapes the and C jk the collisional rate coe cient from level j to k and nl atmosphere, the corresponding energy has been taken out of is the number of energy levels of the atom. These assumptions the thermal pool which corresponds to radiative cooling. In the imply that an excited atom will nearly always fall back to its corona these processes are the dominant ones: collisional excita- ground state through one or more radiative transitions. Hence, tion followed by radiative deexcitation with the photon escaping. photon absorption will always be followed by one or more ra- Collisional de-excitation and absorption of (radiative diative deexcitations, and the gas cannot absorb energy from the excitation) are insignificant in comparison. This is often called radiation field. The only way to lose energy from the gas through the coronal approximation. a bound-bound transition is through a collisional excitation fol- In the chromosphere we have a much more complex situ- lowed by one or more radiative deexcitations. All energy of the ation with all rates playing a role. In strong lines most of the initial excitation is then emitted as radiation and thermal energy photons emitted are immediately absorbed in the same transi- is converted into radiation. tion but the collisional destruction probability is often very low Therefore, as long as Eqs. (2)−(3) hold, the radiative losses and the photon may escape after a very large number of scatter- LXm due to lines of atoms of species X in ionization stage m,per ing events. We may also have photons absorbed in one line (e.g., atom in ionization stage m per electron, is given by Lyman-β) escaping in another transition (e.g., H-α). A full de- scription of these processes entails solving the transfer equations N nl χ C = Xm,1 1 j 1 j , coupled with the non-LTE rate equations (or statistical equilib- LXm (4) N n rium equations if rates are assumed to be instantaneous). Xm j=2 e We here choose to describe the net effect as a combination of three factors that need to be determined empirically from our where NXm,1 is the population density in the ground state of el- simulations: (1) the optically thin radiative loss function (that ement X, ionization stage m, χ1 j is the energy difference be- gives the energy loss from the total generation of local photons tween the ground state (level 1) and level j. Collisions with in the absence of absorption per atom in the ionization stage are dominant under typical chromospheric conditions i ii ii C1 j of consideration (H ,Ca and Mg )), (2) the probability that and and therefore also LX will be a function of temperature ne m this energy escapes from the atmosphere and (3) the fraction of N only (apart from the factor Xm,1 which is close to one). There atoms in the given ionization stage: NXm are therefore reasons to believe that we may write the optically N N thin loss function as a function of temperature only. In Sect. 4.1 = − τ Xm H ρ QX LXm (T)EXm ( ) (T)AX ne (1) NX ρ we will use our numerical simulations to deduce such relations and compare them with the total collisional excitation from the where QX is the radiative heating (positive Q) or cooling (nega- ground state (Eq. (4)). tive Q) per volume from element X (in our case H, Ca and Mg), In the chromosphere not all photons will escape the atmo-

LXm (T) is the optically thin radiative loss function as function of sphere. We parametrize this with the escape probability func- temperature, T, per electron, per particle of element X in ion- tion EX in Eq. (1). In Sect. 4.2 we use our simulations to calcu- τ m ization stage m, EXm ( ) is the escape probability as function of late this escape probability empirically. some depth parameter τ, NXm (T) is the fraction of element X that We complete the recipe by determining in Sect. 4.3 the factor NX NXm NH is in ionization stage m, AX is the abundance of element X, N (T)inEq.(1): the fraction of hydrogen, calcium and magne- ρ X i ii ii is the number of hydrogen particles per gram of stellar material, sium that resides in H ,Ca and Mg . which is a constant for a given set of abundances (=4.407 × 1023 for the solar abundances of Asplund et al. 2009), ne is the elec- tron number density and ρ is the mass density. 4.1. Optically thin radiative loss function The optically thin radiative loss function is equal to the col- lisional excitation rate in the coronal approximation and is then From the RADYN simulation we calculate the total radiative only a function of temperature. In the chromosphere this is not losses as the sum of the net downward radiative rates multi- the case but the hope is that we may still write a radiative loss plied with the energy difference of the transition, summed over function as function of temperature only. We expect various all bound-bound transitions and the Lyman-continuum transition routes between the levels (rather than a collisional excitation in (the other continuum transitions get thin low down in the atmo- one transition followed by a radiative deexcitation in the same sphere and their influence on the energy balance can be taken transition) but as long as collisional de-excitation rates are small into account in a multi-group opacity approach). Figure 2 shows the probability density function (PDF) of the 1 http://cdsweb.u-strasbg.fr/topbase/TheOP.html total radiative losses per neutral hydrogen atom per electron as

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Fig. 2. Probability density function of the radiative losses in lines and Fig. 4. Probability density function of the radiative losses in lines of the Lyman-continuum of H i in the RADYN test atmosphere as function Ca ii as function of temperature in the Bifrost test atmosphere for points of temperature for points above 1.7 Mm height. The curves give the above 1.3 Mm height. The curves give the adopted fit (red) and the total adopted fit (red) and the total collisional excitation according to Eq. (4) collisional excitation according to Eq. (4) (blue). (blue).

Fig. 5. Probability density function of the radiative losses in lines of Fig. 3. Fit to the optically thin cooling per electron per hydrogen particle Mg ii as function of temperature in the Bifrost test atmosphere for points α β (thick solid), contributions from Lyman- (thin black solid), Lyman- above 1.3 Mm height. The curves give the adopted fit (red) and the total γ α (dashed), Lyman- (dot-dashed), H- (red), Lyman-continuum (blue) collisional excitation according to Eq. (4) (blue). and optically thin radiative losses from other elements than hydrogen (thick dotted) as function of temperature.

Lyman-β and the Lyman-continuum are the most important function of temperature for the RADYN simulation2.Weex- ones). At the low temperature end, the Lyman-continuum starts clude the startup phase of the first 1000 s of the simulation and to contribute and H-α eventually dominates below 7 kK. The only include points above 1.7 Mm height to avoid most points hydrogen optically thin radiation dominates over contributions where the escape probability is small. For temperatures above from other elements below 32 kK. 10 kK the relation is very tight and the PDF is close to the total excitation curve. Below 10 kK we have a larger spread, both Figures 4, 5 show the probability density function of the because this part includes points with a low escape probabil- total radiative losses per singly ionized calcium and magne- ity and because the approximations (Eqs. (2)−(3)) start to break sium atom, respectively, as function of temperature for the down. At low temperatures, the in the Lyman tran- Bifrost simulation. This cooling was computed in detail us- sitions builds up quickly when integrating downwards in the at- ing the radiative transfer code Multi3d (Leenaarts & Carlsson mosphere and only the uppermost of these points have an escape 2009) assuming statistical equilibrium and plane-parallel geom- probability close to one. To account for points with low escape etry with each column in the snapshot treated as an independent probability we therefore take the upper envelope as the adopted 1D atmosphere. optically thin radiative loss function. The probability density function is close to the total col- ii ii Figure 3 shows the various contributions to the radiative lisional excitation curve for both Ca and Mg . In the case ii cooling in hydrogen, this time per electron per hydrogen atom of Ca this may be a bit surprising at first glance. The ii 2 (and not per neutral hydrogen atom) to have a normalization Ca 3d D doublet is metastable and we do not include tran- similar to the one used for coronal optically thin radiative losses. sitions from it to the ground state. This implies Eq. (2) does not In most of the temperature range, the Lyman-α transition dom- hold. However, for typical chromospheric electron densities, gas inates with at most 20% coming from other transitions (where temperatures and radiation temperatures in the infrared triplet the following relation holds: 2 This figure, and many of the following figures, are not scatterplots but 2D histograms normalized for each x-axis bin. The darkness at a nl y -axis bin is thus a measure of the probability of finding an atmospheric Rij Cik, j > i. (5) point at this particular y-value given that it is in the x-axis bin. k=1 A39, page 4 of 10 M. Carlsson and J. Leenaarts: Radiative energy balance recipes

This means that an atom in a Ca ii 3d 2D state is much more likely to get radiatively excited to one of the Ca ii 4p 2P states than to move to another level through collisions. Therefore, transition chains starting with collisional excitation from the ground state to a metastable level typically go through a number of radia- tive transitions, and ultimately end in the ground state through a from a Ca ii 4p 2P state. The original col- lision energy is then lost from the gas and Eq. (4) still holds. However, net collisional rates between the excited levels give a larger spread and also radiative losses that sometimes are smaller than the total collisional excitation from the ground state. In con- trast to the case of hydrogen, the spread is thus not caused by optical depth effects and hence we take the average of the values for our fits rather than the upper envelope. At the lowest temper- atures we also get points with radiative heating (not visible in Fig. 6. Probability density function of the empirical escape probability the logarithmic plots) and the fits make a sharp dip downwards. as function of approximate optical depth at Lyman-α line center and the See Sect. 5 for a discussion of radiative heating. adopted fit (red). Only points with a cooling above 5 × 109 erg s−1 g−1 are included and the startup phase of the simulation (first 1000 s) has been excluded. 4.2. Escape probability The assumption that all collisional excitations lead to emit- ted photons breaks down in the chromosphere, where increased mass density leads to increased probability of absorption of pho- tons and collisional deexcitation. Once radiative excitation pro- cesses come into play we have many more possible routes in the statistical equilibrium. To fully account for these processes we have to solve the coupled radiative transfer equations and non-LTE rate equations. Instead, we calculate an empirical es- cape probability from the ratio of the actual cooling as obtained from a detailed radiative transfer computation and the optically thin cooling function determined in Sect. 4.1. Ideally we would like to tabulate this escape probability as a function of optical depth in the lines. Since we are dealing with the total cooling over a number of lines, there is no single optical depth scale to ii ii be chosen. For Ca and Mg we choose to tabulate the escape Fig. 7. Probability density function of the empirical escape probability probability as a function of column mass. The rationale behind as function of column mass for Ca ii and the adopted fit (red). Only this choice is that the optical depth in the resonance lines is pro- points with a a cooling above 5 × 108 erg s−1 g−1 are included. portional to column mass as long as the ionization stage under consideration is the majority species. This is the case for both Ca ii and Mg ii, as shown in Sect. 4.3, in the region of the atmo- as function of the chosen depth-scale. We adopt this ratio as the sphere where the optical depth is significant. Another advantage empirical escape probability as function of depth. with the choice of column mass is that it is easy to calculate from the fundamental variables in a snapshot of an MHD simu- Figure 6 shows the probability density function of the em- ff pirical escape probability as function of approximate optical lation. For hydrogen the situation is di erent. The Lyman lines α get optically thin in the lower part of the transition region where depth at Lyman- line center together with the adopted relation. hydrogen gets ionized. In the 1D simulation used for calculat- Figures 7 and 8 show the probability density function of the em- ffi pirical escape probability as function of column mass for Ca ii ing the empirical coe cients, the transition region stays at a ii constant column mass. On a column mass scale, the hydrogen and Mg together with the adopted relation. escape probability would then be a very steep function. Instead, Note that the adopted relations are not the averages of the − we need to use a depth-variable that is proportional to the neu- points shown in Figs. 6 8 but the ratio between the average of tral hydrogen column density. We use the total column density the actual cooling and the average of the optically thin cool- of neutral hydrogen multiplied with the constant 4 × 10−14 cm2 ing such that points with large cooling get larger weight. The which gives a depth-variable that is close to the optical depth at adopted fit may therefore look like a poor representation of the line center of Lyman-α. The escape probability is high to rather PDFs of Figs. 6–8. If these figures had only included the points high optical depths, because of the very large scattering proba- with the largest cooling, the correspondence would have been bility in the Lyman transitions. more obvious. The escape probability depends in a complicated way on the density, electron density, temperature and local radiation field. 4.3. Ionization fraction We expect it to be close to unity at low column mass density, andtogotozeroatlargecolumnmassdensity. In Sects. 4.1–4.2 we derived expressions for the radiative losses In order to derive an empirical relation we calculate the aver- per atom in the relevant ionization stage per electron. In order age (over time for the RADYN simulation and over space for the to convert these losses to actual losses per volume, the frac- Bifrost snapshot) actual cooling and the average cooling accord- tion of atoms in the ionization stage under study (H i,Caii and ing to the optically thin cooling function determined in Sect. 4.1 Mg ii) must be known. In general this quantity is a function of

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Fig. 10. Probability density function of the fraction of Ca atoms in the Fig. 8. Probability density function of the empirical escape probability form of Ca ii as function of temperature in the Bifrost test atmosphere as function of column mass for Mg ii and the adopted fit (red). Only − − between heights of 0.5 Mm and 2.0 Mm. Curves show the adopted fit points with a a cooling above 5 × 108 erg s 1 g 1 are included. to the PDF (red), the coronal approximation (green) and the coronal approximation with a two-level atom (blue).

Fig. 9. Probability density function of the fraction of neutral H atoms as function of temperature in the RADYN test atmosphere between heights Fig. 11. Probability density function of the fraction of Mg atoms in the of 0.5 Mm and 2.0 Mm after the initial 1000 s. Curves show the adopted form of Mg ii as function of temperature in the Bifrost test atmosphere fit to the PDF (red), and the coronal approximation (blue). between heights of 0.5 Mm and 2.0 Mm. Curves show the adopted fit to the PDF (red), and the coronal approximation (blue).

The best fit curve lies between the coronal approximation curve the electron density, temperature, radiation field, and, in case of for a six-level atom (green curve) and the one for a two-level small transition rates, the history of the atmosphere. atom (blue curve). The reason the coronal approximation for the Neither Saha ionization equilibrium nor coronal equilibrium six-level atom fails is that an overpopulation of the meta-stable are valid in the chromosphere. Therefore no quick physics-based levels drives an upward radiative net rate in the infrared triplet recipe exists that gives the ionization fraction. Fortunately, in our violating the approximation of no upward radiative rates in the 1D and 2D test , the ionization fraction as function coronal approximation. In the limit where this upward rate is of temperature shows a rather small spread around an average much larger than the downward rate we recover the two-level value and an empirical fit to these values gives a reasonable ap- approximation. proximation. We also computed fits as function of temperature Figure 11 shows the ionization balance of Mg ii. The points and electron density. As the ionization fraction is relatively in- are very close to the fitting curve over the full range of the sensitive to the electron density these two-parameter fits do not simulation. yield a significantly better approximation and we settle for rela- tions as function of temperature only. Figure 9 shows the probability density function of the frac- 5. Radiative heating tion of H i, as function of temperature. There is a certain spread In the preceding section we deduced recipes to describe the ra- around the adopted fit which reflects the fact that the hydrogen i ii ii ionzation is also a function of the history of the atmosphere. It is diative cooling in transitions of H ,Ca and Mg . In the cool phases of the chromosphere the atmosphere can be heated by also clear that the coronal approximation gives a rather poor fit radiation in the same transitions. to the simulation data for temperatures below 30 kK. The total radiative heating rate (radiative flux divergence) is Figure 10 shows the same relation for Ca ii. The points are given by rather close to the fitting function except for some points be-  ∞ low 10 kK where an unusually strong radiation field may lead = χ − ν, to a lower fraction of Ca ii for some columns of the atmosphere. Q ν(Jν S ν)d (6) 0 A39, page 6 of 10 M. Carlsson and J. Leenaarts: Radiative energy balance recipes where χν is the opacity, Jν is the mean intensity and S ν is the source function. Assuming a two-level atom and neglecting we have the following relations

NXm,i NXm NH χν = N α φν = α φν A ρ (7) Xm,i 0 0 X ρ NXm NX

S ν = Bν + (1 − )Jν (8)

C ji C ji = ≈ = f (T)ne, (9) A ji + C ji A ji α where NXm,i is the population density of the lower level i, 0 is a constant, φν is the normalized absorption profile of the line, is the collisional destruction probability, Bν is the Planck function, C ji is the collisional deexcitation probability, A ji is the Einstein coefficient for spontaneous emission, f (T) is a function of tem- Fig. 12. Probability density function of the radiative losses in lines of perature and the other terms have the same meaning as in Eq. (1). Ca ii divided by the empirically determined escape probability as func- Inserting into Eq. (6)weget tion of temperature in the Bifrost test atmosphere for points above 0.3 Mm height. The red curve gives the adopted fit.  ∞ NX , N N = α m i φ − ν Xm H ρ. Q f (T) 0 ν(Jν Bν)d AX ρ ne (10) NXm 0 NX

N This equation resembles Eq. (1); Xm,i is a function of tempera- NXm ture in LTE, Jν is constant with height in the optically thin regime  ∞ φν Jν − Bν ν such that 0 ( )d is a function of temperature (but also a function of the source function where Jν thermalizes). At larger optical depths, Jν approaches Bν and the integral behaves simi- larly to the escape probability E(τ)inEq.(1). Although there are many approximations involved, the similarity to Eq. (1) leads us to adopt the same functional form for the heating as we have done for the cooling. We thus just use the previous fits also when LX(T) is negative (corresponding to heating) and adopt for sim- plicity the same escape probability function. To test the applicability of this scheme we plotted the proba- bility density function of the radiative losses (positive) and gains (negative) in the Bifrost test atmosphere, divided by the empiri- Fig. 13. Probability density function of the radiative losses in lines cally defined escape probability, as function of temperature. The ii results are shown in Fig. 12 for CaII and Fig. 13 for MgII. The of Mg divided by the empirically determined escape probability as / function of temperature in the Bifrost test atmosphere for points above figures show that there is a fairly large spread in the losses gains 0.3 Mm height. The red curve gives the adopted fit. at low temperatures. For Ca ii we adopt a fit that gives gains be- low ∼4kK,forMgii we choose zero gains/losses below ∼5kK. Heating in hydrogen transitions is mainly due to absorp- − α Figure 14 shows the cooling ( Q) as a function of column tion of Lyman- photons produced by a nearby source (a strong mass for hydrogen at a time in the RADYN simulation when shock or the transition region). This is illustrated in Fig. 14, α there is a shock in the chromosphere. The detailed cooling is where there is heating due to Ly- photons caused by the emis- dominated by Lyman-α in the shock but with a significant con- 10 m = − . sion at log c 5 2. Such behaviour cannot be modeled with tribution also from H-α. Behind the shock (larger column mass), a simple local equation as Eq. (10) so we choose to set the losses α ∼ H- dominates the cooling. The total cooling is well approxi- and gains for hydrogen to zero below 4kK. mated by our recipe. In front of the shock (smaller column mass) there is heating by absorption of Lyman-α photons. This heating 6. Comparison between detailed solution is not accounted for in the recipe. and recipes Figures 15 and 16 show the average cooling as function of height in the Bifrost test atmosphere. Since hydrogen cooling We have now derived fits for the three factors that enter into the cannot be properly calculated in the Bifrost atmosphere we only radiative cooling: the optically thin radiative loss function, the show the recipe result for hydrogen (and use these values also escape probability and the ionization fraction. To test the accu- for the calculation of the total detailed cooling) in order to show racy with which we can approximate the cooling calculated from the relative importance of the three elements. It is clear that the the detailed solution of the non-LTE rate equations we here com- recipes reproduce the average cooling very well. Hydrogen dom- pare the detailed solution with our recipe. Since the detailed so- inates in the upper chromosphere and transition region while lution from a given test atmosphere has been used to derive the magnesium dominates the cooling in the mid and lower chro- fits we here use different snapshots for this comparison (a dif- mosphere with equally strong calcium cooling at some heights. ferent timeseries calculated with RADYN for hydrogen and a Note that this average cooling varies very much from snapshot snapshot at a much later time calculated with Bifrost for calcium to snapshot depending on the location of shocks in the atmo- and magnesium). sphere. These figures should thus only be used to compare the

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sun and is absorbed in the chromosphere where it contributes to radiative heating. Most of this radiation is absorbed in the con- tinua of neutral helium and neutral hydrogen. If one assumes the coronal ionization equilibrium is only dependent on temper- ature, then the frequency-integrated coronal losses per volume (positive Q means heating, as before) are given by

Qcor = −nH ne f (T). (11)

Here f (T) is a positive function dependent on temperature only that can be pre-computed and tabulated from atomic data. The emissivity associated with this loss is Q η = − cor · (12) 4π Fig. 14. Cooling as function of column mass from hydrogen transitions at t = 3170 s in the RADYN simulation. Total cooling from the detailed Because the coronal radiative losses are integrated over fre- calculation (black solid), from the recipes (black dashed), Lyman-α quency one has to choose a single representative opacity to com- α (blue), H- (red) and the Lyman continuum (green). pute the absorption. A decent choice is to use the opacity at the ionization edge of the ground state of neutral helium (denoted by α). The opacity χ is then given by

NH NHe I χ = α NHe I = αρ AHe , (13) ρ NHe

where NHe I is the number density of neutral helium, NHe is the number density of helium, AHe is the abundance of helium rel- ative to hydrogen, NH is the total number density of hydrogen NH particles and ρ is a constant dependent on the abundances. The quantity NHe I/NHe can be pre-computed from a calibration com- putation as is done for H, Ca and Mg in Sect. 4.3. The most accurate method for computing the absorption of coronal radiation in the chromosphere is to compute the 3D radi- ation field resulting from the coronal emissivity and the absorp- Fig. 15. Average cooling as function of height in the Bifrost test at- tion using the radiative transfer equation: mosphere. Total cooling (black), H i (turquoise), Mg ii (blue) and Ca ii dI (red). According to the detailed calculations (solid) and according to the = η − χI (14) recipes (dashed). ds with I the intensity and s the geometrical path length along a ray. The solution can be computed using standard long or short characteristics techniques in decomposed domains (Heinemann et al. 2006; Hayek et al. 2010). Once the intensity is known the heating rate can be directly computed as 

Qchr = χ I dΩ − η = 4πχJ3D − η, (15) Ω

with Ω the solid angle. The method described above is accurate, but slow. A faster method can be obtained by treating each column in the MHD simulation as a plane-parallel atmosphere and assuming that at each location either η or χ is zero. For each column all quantities then become only a function of height z and Eq. (15) reduces to Fig. 16. As Fig. 15 but for the low-mid chromosphere.  1 Qchr(z) = 2πχ(z) I(z,μ)dμ = 4πχJ1D, (16) detailed solution with the approximate one and not as pictures −1 of the mean chromospheric energy balance. with μ the cosine of the angle with respect to the vertical. If we furthermore assume the emitting region is located on 7. Heating from incident radiation field top of the absorbing region, we can ignore rays that point out- ward from the interior of the in the integral of Eq. (16). Part of the optically thin radiative losses from the corona escapes The intensity that impinges on the top of the absorbing region outwards, but an equal amount of energy is emitted towards the for inward pointing rays is simply the integral of the emissivity

A39, page 8 of 10 M. Carlsson and J. Leenaarts: Radiative energy balance recipes along the z-axis from the top (zt) to the bottom (zb)ofthecom- putational domain, weighted with the inverse of μ to account for slanted rays  zb μ = 1 η . I( )cor |μ| (z)dz (17) zt

Now that we have computed the incoming intensity that im- pinges on the top of the absorbing region we can compute the intensity at all heights from the solution to the transfer equation with zero emissivity:

−τ(z)/|μi| I(z,μ) = Icor(μ)exp . (18)

The vertical optical depth τ is computed from the top of the com- putational domain:  z τ(z) = χ(z)dz. (19) zt

The integrals in Eqs. (17)and(19) can both be computed effi- ciently in parallel and need to be computed only once, indepen- dent of the number of μ-values in the angle discretization. The method gives inaccurate results if the assumption of coronal emission above chromospheric absorption fails. This happens for example when a cool finger of chromospheric gas protrudes slanted into the corona. Figure 17 shows a comparison of the 3D and the fast 1D method. The top panel shows the coronal emissivity η.It is mostly smooth with occasional very large peaks close to the transition region (red curve). The emissivity for gas colder than 30 kK (below the red curve) is negligible. The second panel shows the angle-averaged radiation field computed in 3D (J3D in Eq. (15)). The radiation field is fairly smooth as it is determined by an average of the emissivity throughout the corona. The middle panel shows the angle-averaged radiation field treating each column as a plane-parallel atmosphere (J1D in Eq. (16)). Here it is far from smooth. The radiation field is high in columns with a large integrated emissivity, as indicated by the Fig. 17. Comparison of the 1D and 3D method for computing chromo- green and red vertical stripes. Note that there is cool gas located spheric heating due to absorption of coronal radiation. All panels show a above coronal gas around (x, z) = (8, 3) Mm. Here the radiation vertical slice through a snapshot of a three-dimensional radiation-MHD field is highest at the top and decreases downward, while the simulation performed with Bifrost. The red curve is located at a gas tem- emissivity panel shows that the emission is mainly coming from perature of 30 000 K and indicates the location of the transition zone. the coronal gas located below the cool finger. All brightness ranges have been clipped to bring out variation at low values of the displayed quantities. Panels, from top to bottom: coro- The fourth panel shows the 3D heating rate. The heating rate η is non-zero only below the red curve, validating our assumption nal emissivity ; angle-averaged radiation field computed in 3D; angle- averaged radiation field computed in 1D: heating rate per mass with for the fast 1D method of separate absorbing and emitting re- 3D radiation: heating rate per mass with 1D radiation. Note that the gions. It is fairly smooth, with quite some heating in the cool 3D radiation field is computed using the full 3D volume, not only this bubble around (x, z) = (4, 4) Mm, caused by coronal radiation 2D slice. coming in from the sides. The bottom panel shows the 1D heat- ing rate. It shows large extrema in columns with large integrated emissivity, and is less smooth than the 3D heating. Because it 8. Conclusions is computed column-by-column, it cannot include the sideways radiation in the cool bubble as indicated by the low heating rate We have shown that the radiative cooling in the solar chromo- there compared to the 3D case. However, its overall appearance sphere by hydrogen lines, the Lyman continuum, the H and K is quite similar to the 3D heating rate. and infrared triplet lines of singly ionized calcium, and the h and Figure 18 compares the horizontally averaged heating rate k lines of singly ionized magnesium can be expressed fairly ac- as function of height computed from the 1D recipe with the curately as easily computed functions. These functions are the 3D heating rate. The recipe is always within 15% of the 3D heat- product of an optically thin emission (dependent on tempera- ing. Part of this difference is caused by numerical errors because ture), an escape probability (dependent on column mass) and the strong coronal emission close to the transition region is only an ionization fraction (dependent on temperature). While at any barely resolved on the grid used in the simulation. given location in the atmosphere our recipe might deviate from

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